of course, for a general transformation [itex]\Lambda[/itex], [itex]\eta_{uv}[/itex] might not turn out to be the same after the transform as it did for this example, you have to assume that the transform preserves the lengths of space-time intervals to make this happen.

ps a minor Latex point - note that Lambda^a{}_b gives [tex]\Lambda^a{}_b[/tex], properly aligned, as opposed to Lambda^a_b which gives [tex]\Lambda^a_b[/tex]

What sort of transformation is [itex]\Lambda[/itex] supposed to be? It looks like my boost example, for instance, does not have [itex]\Lambda[/itex] in this form for large beta, because the diagonal coefficients are 1/sqrt(1-beta^2), not unity.